# Fast algorithms for computing the distance to instability of nonlinear eigenvalue problems, with application to time-delay systems

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## Abstract

A continuous dynamical system is stable if all eigenvalues lie strictly in the left half of the complex plane. However, this is not a robust measure because stability is no longer guaranteed when the system parameters are slightly perturbed. Therefore, the pseudospectrum of a matrix and its pseudospectral abscissa are studied. Mostly, one is often interested in computing the distance to instability, because it is a robust measure for stability against perturbations. As a first contribution, this paper presents two iterative methods for computing the distance to instability, considering complex perturbations. The first one is based on locating a zero of the pseudospectral abscissa function. This method is particularly suitable for large sparse matrices as it is based on repeated eigenvalue computations, where the original matrix is perturbed with a rank one matrix. The second method is based on a recently proposed global optimization technique. The advantages of both methods can be combined in a hybrid algorithm. As a second contribution we show that the methods apply to a broad class of nonlinear eigenvalue problems, in particular eigenvalue problems inferred from linear delay-differential equations, and, therefore, they are useful for a wide range of problems. In the numerical examples the standard eigenvalue problem, the quadratic eigenvalue problem and the delay eigenvalue problem are addressed.

## Keywords

Spectral abscissa Stability Nonlinear eigenvalue problems Pseudospectrum Pseudospectral abscissa Distance to instability## 1 Introduction

*spectral abscissa*of \(A\), which is the maximum of the real parts of the eigenvalues, is negative. Unfortunately, the spectral abscissa is not a robust measure for stability. This paper focuses on the effects of perturbations on the stability of a given dynamical system. Therefore the \(\varepsilon \)-pseudospectrum \(\varLambda _{\varepsilon }\left( A\right) \) is studied, i.e. the set of the eigenvalues of all real or complex matrices in a given neighborhood of \(A\). In particular,

*pseudospectral abscissa*\(\alpha _\varepsilon \left( A\right) \), i.e.,

## 2 Preliminaries

### 2.1 Pseudospectra and distance to instability for nonlinear eigenvalue problems

*complex*matrices, i.e.,

### **Proposition 1**

We say that \(F\) is exponentially stable if all zeros are confined to the open left half plane and bounded away from the imaginary axis, i.e., \(\alpha (F)<0\). To assess the robustness of stability w.r.t. perturbations on the coefficient matrices \(A_i\) we introduce the concept of distance to instability.

### **Definition 1**

To conclude the section we state and discuss the main assumption made throughout the paper.

### **Assumption 1**

There exist a number \(\varepsilon _{\max }>\mu (F)\) such that for arbitrary but fixed \(r\in \mathbb {R}\) and \(\varepsilon \in [0,\ \varepsilon _{\max })\) the set \(\varLambda _{\varepsilon }\cap \{\lambda \in \mathbb {C}:\ \mathrm{Re }\left( \lambda \right) \ge r\}\) is bounded.

For delay eigenvalue problem (15), Assumption 1 is satisfied for any value of \(\varepsilon _{\max }\) larger than \(\mu (F)\), as long as only \(A_1\) and \(A_2\) are perturbed. For quadratic eigenvalue problem (14) with nonsingular \(A_2\), the assumption is satisfied by taking \(\varepsilon _{\max }=w_2 \sigma _n(A_2)\), with \(w_2\) the weight on the perturbation of \(A_2\). This bound stems from the fact that for \(\varepsilon >w_2\sigma _n(A_2)\) the \(\varepsilon \)-pseudospectrum is unbounded in all directions in the complex plane, containing perturbations of an eigenvalue at infinity [7, Section 2.2].

Assumption 1 implies that, by varying \(\varepsilon \) in a continuous way, a transition from the situation where \(\alpha _{\varepsilon }<0\) to a situation where \(\alpha _{\varepsilon }\ge 0\) is characterized by eigenvalues moving from the open left half plane to the imaginary axis (i.e., right half plane eigenvalues coming from the point at infinity cannot occur). Combined with the characterization (12) this leads to the following expression for the stability radius [10, Corollary 3].

### **Proposition 2**

## 3 Previous work

The approach proposed in [7] to compute the \(\varepsilon \)-pseudospectral abscissa for a nonlinear eigenvalue problem is inspired by [6] where an iterative algorithm is proposed to find a locally rightmost point of the pseudospectrum of a matrix. This algorithm is mainly based on the property that a boundary point of the pseudospectrum is an eigenvalue of \(A + \varepsilon uv^*\), with \(u\) and \(v\) respectively left and right corresponding eigenvectors. This gives rise to a fixed-point iteration method where the left and right eigenvectors determine the next perturbation. In [12] a continuous dynamical version of the algorithm is presented. More precisely, a differential equation on the manifold of normalized rank one matrices is constructed, having as stationary point an optimal perturbation for which the corresponding rightmost eigenvalue is an extremal point of the pseudospectrum. Extensions to real perturbations are described in [13].

### 3.1 Computing pseudospectral abscissa for large-scale linear and nonlinear eigenvalue problems

In what follows, we review the method of [7], which generalizes the method of [6] to nonlinear eigenvalue problems.

### **Assumption 2**

- 1.
the smallest singular value of \(\sum _{i=0}^m A_i p_i(\lambda _{\varepsilon })\) is simple;

- 2.the rightmost eigenvalue ofis simple;$$\begin{aligned} \left( \sum _{i=0}^m \left( A_i-\frac{\overline{p_i(\lambda _{\varepsilon })}}{w_i |p_i(\lambda _{\varepsilon })|} u_{\varepsilon } v_{\varepsilon }^* \varepsilon \right) p_i(\lambda )\right) v=0 \end{aligned}$$(17)
- 3.
we have \(p_i(\lambda _{\varepsilon })\ne 0\) whenever \(w_i\) is finite, for \(0\le i\le m\).

Refinements of the basic algorithm to enforce global convergence to the globally rightmost point of the pseudospectrum can be found in [7], consisting of an adaptation to make the sequence \(\{\mathrm{Re }\left( \lambda _k\right) \}_{k\ge 1}\) monotonic (and in this way preventing, for example, that the iterations converge to a non-equilibrium solution), and the possibility to start with several rightmost eigenvalues (to avoid converging to a locally but not globally rightmost point). All variants share the property that their application only requires methods to compute the *rightmost* eigenvalues and the corresponding left and right eigenvectors, for which fast iterative solvers can be used if the system matrices are large and sparse. This feature makes them also applicable to large-scale problems. The local convergence is linear, with a convergence factor tending to zero as \(\varepsilon \rightarrow 0\). For an extended discussion on this point see [7].

An application of the method to compute the distance to instability has been considered in [14] and is described in Sect. 4.1.

## 4 Distance to instability: local method based on root finding of spectral abscissa function

### 4.1 Newton’s method

*pseudospectral abscissa function*

### **Lemma 1**

The function (25) is continuous, increasing, and, if not all weights are infinity, then we have \(\lim _{\varepsilon \rightarrow \infty } \alpha _{\varepsilon }=\infty .\)

### **Theorem 3**

^{1}then we have

The proof can be found in [14]. Note that the differentiability of \(\lambda \) in a neighbourhood of \(\hat{\varepsilon }\) follows from the implicit function theorem applied to (18)–(21).

### 4.2 The novel method: mixing the inner and outer iteration loop

Newton’s iteration (28) is characterized by two nested loops. In the outer iteration loop, the value of \(\varepsilon \) is updated. Intrinsic to Newton’s method, the convergence is quadratic in the generic case. In an inner iteration loop, the pseudospectral abscissa is computed for a fixed value of \(\varepsilon \) using the linearly converging Algorithm 1. In order to decrease the computational cost, one may argue that an accurate computation of the pseudospectral abscissa is not needed in the first outer iterations, but only when \(\varepsilon \) is close to \(\mu (F)\). One can go a step further and replace the computation of the pseudospectral abscissa by one iteration step of Algorithm 1, and update \(\varepsilon \) based on the current value of \(\lambda _k\), as if it had already converged to the globally rightmost point. This idea lies at the basis of Algorithm 2. We call the underlying method the *method of adaptive perturbations* because in every iteration step both the value of \(\varepsilon \) and the perturbations on the coefficient matrices are updated. Note that a fixed point \((\lambda ,\varepsilon ,u,v)\) of the iteration satisfies the conditions (18)–(21), and in addition we have \(\mathrm{Re }\left( \lambda \right) =0\).

It should be noted that in case condition (30) and (31) do not allow a unique solution, the same remedies explained for Algorithm 1 apply. This situation has, however, not occurred in our experiments.

In all our experiments, represented in Sect. 5, Algorithm 2 turns out to be significantly faster then the iteration (28), while loss of convergence due to a nesting of iterations has not been observed. We note that, also with the update of \(\varepsilon \), the method only relies on the computation of the rightmost eigenvalue of perturbed problems, characterized by rank-one updates on the original system matrices. The latter property can be exploited by iterative solvers, both in matrix vector products and in solving linear systems, where the Sherman–Morrison–Woodbury formula plays a key-role. The overall cost of Algorithm 2 is comparable to the cost of Algorithm 1.

## 5 Distance to instability via global optimization

- 1.A routine which, for a given value of \(\omega \), returns the objective function and its derivative whenever it is differentiable. The computation of \(f\) mainly amounts to computing the smallest singular value of matrix \(F(j\omega )\). Hence, a specific solver for nonlinear eigenvalue problems is not required, in contrast to the methods proposed in the previous section. In the generic case where \(p_i(j\omega )\ne 0, i=0,\ldots m\) and, in addition, the smallest singular value of \(F(j\omega )\) is simple, the derivative \(f'(\omega )\) exists and we can expresswhere \(u\) and \(v\) are normalized left and right singular vectors corresponding to the smallest singular value of \(F(j\omega )\).$$\begin{aligned} f'(\omega )&= -\,\frac{ \mathrm{Im }\left( u^* \left( \sum _{i=0}^m A_i p_i'(j\omega )\right) v\right) }{\sum _{i=0}^m \frac{|p_i(j\omega )|}{w_i}}\\&+\,\frac{\sigma _n\left( \sum _{i=0}^m A_i p_i\left( j\omega \right) \right) \sum _{i=0}^m \frac{\mathrm{Im }\left( p_i(-j\omega )p_i'(j\omega )\right) }{w_i|p_i(j\omega )|} }{\left( \sum _{i=0}^m \frac{|p_i(j\omega )|}{w_i}\right) ^2 }, \end{aligned}$$
- 2.
Prior knowledge of a compact interval which contains the global minimizer over \(\mathbb {R}\). For the delay eigenvalue problem such an interval can be computed as in [15], but the obtained bounds may be conservative. Almost always the heuristic choice \([0,\ \omega _{m}]\) is sufficient, where \(\omega _m=1.2\ \mathrm{Im }\left( \lambda _m\right) \), with \(\lambda _m\) the dominant eigenvalue with largest imaginary part. A practical choice is to consider three dominant eigenvalues.

- 3.
A lower bound \(-\gamma \) on the second derivative of the scaled singular value curves, holding over the whole interval under consideration. It is impractical to compute the second derivative and therefore, in our implementation, a piece-wise cubic approximating function of \(f\) over the interval \([0,\omega _m]\) is used to estimate a bound.

## 6 Numerical experiments

Problem | Method | \(\varepsilon _d\) | \(\alpha _{\varepsilon _d}\) | # iter | Time (s) |
---|---|---|---|---|---|

airy | Newton | 4.814833246948e\(-\)02 | 2.92e\(-\)11 | 22 | 392.1 |

(\(n=99\)) | Adaptive | 4.814833245230e\(-\)02 | 6.42e\(-\)12 | 114 | 1.77 |

Optimization | 4.814833244747e\(-\)02 | 7.44e\(-\)15 | 127 | 0.87 | |

transient | Newton | 5.402011219865e\(-\)10 | \(-\)5.75e\(-\)09 | 10 | 1.02 |

(\(n=100\)) | Adaptive | 5.402091562084e\(-\)10 | \(-\)9.88e\(-\)08 | 10 | 0.24 |

Optimization | 5.402008418433e\(-\)10 | \(-\)1.42e\(-\)08 | 34874 | 197.3 | |

lshape | Newton | 9.108293989259e\(-\)03 | 2.33e\(-\)15 | 1 | 3.27 |

(\(n=3466\), sparse) | Adaptive | 9.108293989257e\(-\)03 | 6.21e\(-\)16 | 3 | 3.17 |

Optimization | 9.108293989353e\(-\)03 | 9.67e\(-\)14 | 2 | 0.19 |

Quadratic eigenvalue problems [23]: distance to instability \(\varepsilon _d\) and corresponding pseudospectral abscissa \(\alpha _{\varepsilon _d}\) computed with Newton’s method, the adaptive perturbation method, the optimization method, and a hybrid version of the previous two methods

Problem | Method | \(\varepsilon _d\) | \(\alpha _{\varepsilon _d}\) | # iter | Time (s) |
---|---|---|---|---|---|

hospital | Newton | 8.139884576182e-02 | 0.58 | 3 | 0.09 |

\((n=24)\) | Adaptive | 8.139884576180e-02 | 0.58 | 8 | 0.05 |

Optimization | 4.400202122331e\(-\)02 | 1.33e\(-\)14 | 1067 | 25.00 | |

Hybrid | 4.400202122334e\(-\)02 | 5.37e\(-\)13 | 803/8 | 14.66 | |

pdde_stability | Newton | 9.231159788143e\(-\)01 | \(-\)3.38e\(-\)14 | 5 | 78.88 |

(\(n=225\), sparse) | Adaptive | 9.231159788137e\(-\)01 | \(-\)2.94e\(-\)13 | 35 | 40.55 |

Optimization | 9.231159788142e\(-\)01 | \(-\)6.55e\(-\)14 | 44 | 1.16 | |

Hybrid | 9.231159788148e\(-\)01 | 3.40e\(-\)13 | 13/23 | 30.22 | |

sign2 | Newton | 6.686158451176e\(-\)01 | 4.54e\(-\)14 | 13 | 552.10 |

\((n=81)\) | Adaptive | 6.686158451149e\(-\)01 | 8.50e\(-\)13 | 467 | 44.46 |

Optimization | 6.686158450798e\(-\)01 | \(-\)5.81e\(-\)11 | 88 | 0.66 | |

Hybrid | 6.686158450920e\(-\)01 | 8.36e\(-\)13 | 20/67 | 6.50 |

For computing the distance to instability of a nonlinear eigenvalue problem with one of the root finding methods (Newton’s iteration (28) and Algorithm 2), we need a solver for computing the rightmost eigenvalue \(\lambda _{\mathrm {RM}}\). In the recent literature [16, 17, 18, 19], there exist several general nonlinear eigenvalue solvers, which are not only applicable to small problems but also to large sparse problems. In these numerical experiments the quadratic eigenvalue problem is first linearized and then \(\lambda _{\mathrm {RM}}\) can be easily computed using eig or eigs. For computing \(\lambda _{\mathrm {RM}}\) of the delay problems we used the delay eigenvalue solver proposed in [20]. On the other hand, the method based on global optimization (Sect. 5) for computing the distance to instability of a nonlinear eigenvalue problem only requires the computation of the smallest singular value of a constant matrix. For this we used svd or svds.

### 6.1 Linear eigenvalue problems

For the experiments on linear eigenvalue problems we take two examples (‘airy’ and ‘transient’) of the EIGTOOL collection [21] and one example (‘lshape’) of the Harwell–Boeing collection [22]. This last example is shifted with shift \(\sigma = 7\) in order to stabilize the matrix. The results of the computations of the distance to instability for these linear eigenvalue problems are displayed in Table 1.

Delay eigenvalue problems: distance to instability \(\varepsilon _d\) and corresponding pseudospectral abscissa \(\alpha _{\varepsilon _d}\) computed with Newton’s method, the adaptive perturbation method, the optimization method, and a hybrid version of the previous two methods

Problem | Method | \(\varepsilon _d\) | \(\alpha _{\varepsilon _d}\) | # iter | Time (s) |
---|---|---|---|---|---|

small-scale | Newton | 1.762769038185791 | \(-\)7.46e-16 | 5 | 4.79 |

(\(n=2\)) | Adaptive | 1.762769038190436 | 8.86e\(-\)13 | 37 | 0.80 |

Optimization | 1.762769038184915 | \(-\)3.21e\(-\)13 | 53 | 0.15 | |

Hybrid | 1.762769038189088 | 6.73e\(-\)13 | 26/35 | 0.78 | |

large-scale | Newton | 0.249999991774308 | 4.58e\(-\)15 | 5 | 75.93 |

(\(n=5000\), sparse) | Adaptive | 0.249999991774340 | 4.19e\(-\)14 | 8 | 6.60 |

Optimization | 0.249999991776921 | 3.27e\(-\)12 | 30 | 18.68 | |

Hybrid | 0.249999991774678 | 4.60e\(-\)13 | 15/5 | 13.57 |

### 6.2 Polynomial eigenvalue problems

For the experiments on polynomial eigenvalue problems we take three (shifted) examples (‘hospital’, ‘pdde_ stability’ and ‘sign2’) of the NLEVP collection [23]. We applied perturbations to all system matrices and took unity weights, i.e., \(w_i = 1\). The results of the computations are displayed in Table 2.

The ‘hospital’ problem shows that the root finding methods (Newton’s iteration (28) and Algorithm 2) converge to a locally rightmost point, whereas the method based on optimization (Sect. 5) converges to the globally rightmost point. However, to achieve full accuracy of the global minimum, the optimization method requires a lot of iterations, since for this problem we needed to take to into account ten dominant eigenvalues. Therefore, a wise combination of the proposed Algorithms 2 and the global optimization method can tackle this limitation.

In particular, first execute the optimization method with high tolerance. If the lower bound is good, this can be done very fast. Assume the minimizer \(\tilde{\omega }\) is found. Then the point \(j\tilde{\omega }\) lies on the boundary of the \(\varepsilon \)-pseudospectrum where \(\varepsilon \) equals the objective function value. The corresponding perturbation can be constructed from the singular vectors by using [6, Lemma 1.1] for linear problems and [7, Proposition 3.1] for nonlinear problems. Subsequently, adaptive perturbations can be constructed by starting from \(\lambda _0 = j\tilde{\omega }\), and \(\varepsilon \) equal to the objective function value and the aforementioned perturbation. This almost always results in a global solution with high accuracy, see, e.g., Table 2 where we set the required tolerance to 1e\(-\)12.

### 6.3 Delay eigenvalue problems

## 7 Concluding remarks

Two algorithms have been adapted, combined and implemented for computing the distance to instability, for both linear and nonlinear eigenvalue problems. The two algorithms are well suited for large-scale sparse problems. Although for the method with adaptive perturbations only convergence to a locally rightmost point of the pseudospectrum on the imaginary axis can be guaranteed, a high accuracy can be achieved. The second method directly solves an optimization problem inferred from the characterization (12). As the main advantage, the global optimum can be found, yet the convergence rate can be slow if the second derivative of the objective function cannot be accurately estimated. Therefore, a hybrid algorithm is recommended which almost always converges globally with high accuracy and a reasonable computation time.

Recently, several algorithms have been proposed for computing extremal points of *structured* and / or *real* pseudospectra of matrices, for the Euclidean and the Frobenius norm (see, e.g., [13] for real pseudospectra). This class of methods is based on either a discrete iteration or on a differential equation on a manifold of low rank matrices, which originate from the property that in the cases under consideration the boundary of the pseudospectra can be reached by applying low rank perturbations to the matrix. It is expected that the idea of updating \(\varepsilon \), behind Algorithm 2, applies to all these methods, which, in this way, could be extended to the computation of the corresponding distances to instability. Extending the presented approach based on global optimization to structured or real perturbations seems more difficult. For the standard eigenvalue problem, an extension to real perturbations measured with the Euclidean norm is still possible. The key is to replace the expression for the distance to instability (12) by a characterization in terms of real structured singular values [24, 25]. However, computing real structured singular values is significantly more demanding computationally than computing (standard) singular values.

## Footnotes

- 1.
As we assume that the spectrum is symmetric w.r.t. the real axis, only the eigenvalues in the upper half plane are considered.

## Notes

### Acknowledgments

The authors wish to thank the anonymous referees for their helpful comments and suggestions that improved the quality of the paper. This work was supported by the Programme of Interuniversity Attraction Poles of the Belgian Federal Science Policy Oce (IAP P6-DYSCO), by OPTEC, the Optimization in Engineering Center of the KU Leuven, by projects STRT1-09/33 and OT/10/038 of the KU Leuven Research Council, and by project G.0712.11N of the Research Foundation-Flanders (FWO).

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